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Answer :
We start with the expression
[tex]$$
x^6 + 125x^3.
$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that both terms contain a factor of [tex]$x^3$[/tex]. Factor [tex]$x^3$[/tex] out:
[tex]$$
x^6 + 125x^3 = x^3\left(x^3 + 125\right).
$$[/tex]
Step 2. Recognize the sum of cubes.
The expression inside the parentheses, [tex]$x^3 + 125$[/tex], is a sum of cubes because [tex]$125$[/tex] can be written as [tex]$5^3$[/tex]. Recall the sum of cubes formula:
[tex]$$
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right).
$$[/tex]
Here, let [tex]$a = x$[/tex] and [tex]$b = 5$[/tex]. Therefore,
[tex]$$
x^3 + 125 = (x + 5)\left(x^2 - 5x + 25\right).
$$[/tex]
Step 3. Combine the factors.
Substitute the factored form of [tex]$x^3 + 125$[/tex] back into the expression:
[tex]$$
x^6 + 125x^3 = x^3 (x + 5) \left(x^2 - 5x + 25\right).
$$[/tex]
Step 4. Rewrite the factorization to match one of the given choices.
Notice that we can group the factors as follows:
[tex]$$
x^3 (x + 5) \left(x^2 - 5x + 25\right)
= \Big[x \cdot x^2 (x + 5) \Big] \cdot \left(x^2 - 5x + 25\right).
$$[/tex]
Rearrange the factors by writing [tex]$x^3$[/tex] as [tex]$x \cdot x^2$[/tex]. Group [tex]$x$[/tex] with [tex]$(x+5)$[/tex]:
[tex]$$
x^3 (x + 5)\left(x^2 - 5x + 25\right) = \Big[x(x + 5)\Big]\left[x^2\left(x^2 - 5x + 25\right)\right].
$$[/tex]
This gives us:
- First factor: [tex]$x(x+5) = x^2 + 5x$[/tex],
- Second factor: [tex]$x^2\left(x^2 - 5x + 25\right) = x^4 - 5x^3 + 25x^2$[/tex].
Thus, the complete factorization can be written as:
[tex]$$
x^6 + 125x^3 = \left(x^2+5x\right)\left(x^4-5x^3+25x^2\right).
$$[/tex]
This matches option (a).
[tex]$$
x^6 + 125x^3.
$$[/tex]
Step 1. Factor out the greatest common factor.
Notice that both terms contain a factor of [tex]$x^3$[/tex]. Factor [tex]$x^3$[/tex] out:
[tex]$$
x^6 + 125x^3 = x^3\left(x^3 + 125\right).
$$[/tex]
Step 2. Recognize the sum of cubes.
The expression inside the parentheses, [tex]$x^3 + 125$[/tex], is a sum of cubes because [tex]$125$[/tex] can be written as [tex]$5^3$[/tex]. Recall the sum of cubes formula:
[tex]$$
a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right).
$$[/tex]
Here, let [tex]$a = x$[/tex] and [tex]$b = 5$[/tex]. Therefore,
[tex]$$
x^3 + 125 = (x + 5)\left(x^2 - 5x + 25\right).
$$[/tex]
Step 3. Combine the factors.
Substitute the factored form of [tex]$x^3 + 125$[/tex] back into the expression:
[tex]$$
x^6 + 125x^3 = x^3 (x + 5) \left(x^2 - 5x + 25\right).
$$[/tex]
Step 4. Rewrite the factorization to match one of the given choices.
Notice that we can group the factors as follows:
[tex]$$
x^3 (x + 5) \left(x^2 - 5x + 25\right)
= \Big[x \cdot x^2 (x + 5) \Big] \cdot \left(x^2 - 5x + 25\right).
$$[/tex]
Rearrange the factors by writing [tex]$x^3$[/tex] as [tex]$x \cdot x^2$[/tex]. Group [tex]$x$[/tex] with [tex]$(x+5)$[/tex]:
[tex]$$
x^3 (x + 5)\left(x^2 - 5x + 25\right) = \Big[x(x + 5)\Big]\left[x^2\left(x^2 - 5x + 25\right)\right].
$$[/tex]
This gives us:
- First factor: [tex]$x(x+5) = x^2 + 5x$[/tex],
- Second factor: [tex]$x^2\left(x^2 - 5x + 25\right) = x^4 - 5x^3 + 25x^2$[/tex].
Thus, the complete factorization can be written as:
[tex]$$
x^6 + 125x^3 = \left(x^2+5x\right)\left(x^4-5x^3+25x^2\right).
$$[/tex]
This matches option (a).
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